$\bigcup_{n \in \mathbb{N}} \bigcup_{u_1<\ldots<u_n \leq t} \sigma(X_{u_1},\ldots,X_{u_n})$ is a $\cap$-stable system

Question

I would like to show, that given a stochastic process $(X_u)_{u\in\mathbb{R}_{\ge0}}$, that $\mathcal{D} := \bigcup_{n \in \mathbb{N}} \bigcup_{u_1<\ldots<u_n \leq t} \sigma(X_{u_1},\ldots,X_{u_n})$ is a $\cap$-stable system. However, it seems to me that a generic element of $\sigma(X_{u_1},\ldots,X_{u_n})$ is written $(X_{u_1},\ldots,X_{u_n})^{-1}(B)$ with $B\in\mathcal{B}(\mathbb{R}^n)$. I know that $\mathcal{B}(\mathbb{R}^n)=\sigma(\mathcal{B}(\mathbb{R})\times\ldots\times\mathcal{B}(\mathbb{R}))$ and that $(X_{u_1},\ldots,X_{u_n})^{-1}(B_1\times\ldots\times B_n)\cap (X_{v_1},\ldots,X_{v_m})^{-1}(C_1\times\ldots\times C_m)\in\mathcal{D}$ but I am not sure how to conclude.

Answer 1

However, it seems to me that a generic element of $\sigma(X_{u_1},\ldots,X_{u_n})$ is written $(X_{u_1},\ldots,X_{u_n})^{-1}(B)$ with $B\in\mathcal{B}(\mathbb{R}^n)$.

That is true, but you don't need it here.

Suppose $A,B \in \mathcal{D}$. Then, by definition of union, there exists some integer $n$ and some times $u_1, \dots, u_n$ such that $A \in \sigma(X_{u_1}, \dots, X_{u_n})$. Likewise, there exists $m$ and times $v_1, \dots, v_m$ such that $B \in \sigma(X_{v_1}, \dots, X_{v_m})$.

So $A$ and $B$ are both in $\sigma(X_{u_1}, \dots, X_{u_n}, X_{v_1}, \dots, X_{v_m})$. The latter is a $\sigma$-field so $A \cap B \in \sigma(X_{u_1}, \dots, X_{u_n}, X_{v_1}, \dots, X_{v_m})$. And $\sigma(X_{u_1}, \dots, X_{u_n}, X_{v_1}, \dots, X_{v_m})$ is one of the $\sigma$-fields whose union is $\mathcal{D}$, so $A \cap B \in \mathcal{D}$.